Modal logic…part deux

Background

In a previous blog post (Durvasula 2026), I had argued that inferring category structure directly from phonetic measurements was fraught. It was particularly problematic in exactly the cases where folks typically think fine-grained phonetic evidence helps, namely, in the case where the two categories are quite overlapped. The fundamental issue is that a mixture population of two underlying distributions has a single mode once the categories are sufficiently overlapped, and is indistinguishable from a single distribution, when

\[\begin{equation} |Mean_1 - Mean_2| ≤ 2*min(σ_1, σ_2) \end{equation}\]

The above threshold means that no inferential statistical test will help you once the difference in means is below that threshold. that the actual mixture distribution is neutral on the nature of the underlying componentry [Edit on 30 Oct 2026: I want to be careful here. One can still have an inference with respect to the number of components and what they are, and it can even be consistent. But that inference is based on a bias in the inference procedure related to the number of categories; for example, some sort of a prior, that deviates from equiprobability, on whether there are one or two categories. Different algorithms with different degrees of bias on the matter, will then lead to potentially different inferences.] More generally, as Murphy (1964) correctly put it ``It is one thing to argue from mechanisms to expected outcomes; it is very much more difficult and hazardous to argue from observations back to mechanisms.’’ (p. 312).

At the end of that post, I mentioned that a student of mine, Huanbin Teng (now at Stony Brook University) was looking at the more general issue of category overlap and common statistical tests. We have just submitted a manuscript for review — it includes mathematical results, simulations and testing of different statistical techniques, and finally testing on real distributions. Here it is:

Everything that quacks ain’t a duck. What can we learn from phonetic distributions?
Huanbin Teng and Karthik Durvasula
Phonetic studies sometimes infer how many categories a linguistic system contains using surface distributional diagnostics. This inference is examined by distinguishing four levels: distribution shape, labeled-group separation, statistical components, and linguistic categories. In an equal-proportion, equal-variance Gaussian mixture, two speech categories produce a unimodal pooled distribution when the distance between their means is less than two standard deviations. In many realistic contexts, unimodality of a pooled distribution alone does not establish the presence of a single linguistic category. Conversely, bimodality of a pooled distribution alone does not establish two linguistic categories. The standard mixture-variance decomposition also shows that a two-category mixture need not have greater variance than its more variable component.Simulations show mode-based diagnostics can correctly describe surface modality yet poorly recover the categories that generated the data. Empirical cases further illustrate that distribution shape, labeled-group separation, and component structure capture distinct properties of the data and need not converge on the same category analysis.These diagnostics should thus be interpreted at their own levels. Agreement among them can strengthen a description of surface structure, but they cannot determine category hood on their own. Category-level claims require converging linguistic evidence suited to the category under study.